Or even, and simpler, just compare it to [tex]\frac{1}{n^2}[/tex]. As soon as n> 2,
[tex]\frac{1}{n^n}< \frac{1}{n^2}[/tex]. Now, what do you know about the convergence of [tex]\Sigma_1^\infty\frac{1}{n^2}[/tex]?